Show, by multiplying the matrices, that the following equation represents an ellipse. $$\left(\begin{array}{ll}x & y\end{array}\right)\left(\begin{array}{rr}5 & -7 \\\7 & 3 \end{array}\right)\left(\begin{array}{l}x \\\y\end{array}\right)=30.$$

Matrix multiplication is a fundamental algebraic operation used extensively in various mathematical disciplines. To multiply two matrices, align and pair their rows and columns. In the problem, we began by identifying the matrices involved in the equation: \( \begin{array}{ll}x & y\right)\) and \( \begin{array}{rr}5 & -7\7 & 3\right)\).

The process entails multiplying each element of the row by the corresponding element in the column and summing these products. When we multiplied these matrices, the intermediate result was: \( \begin{array}{c}5x - 7y\7x + 3y\right)\). Finally, the resulting vector was multiplied by \( \begin{array}{ll}x & y\right)\), yielding: \( (x)(5x - 7y) + (y)(7x + 3y) = 5x^2 + 3y^2\), which we set equal to 30. This method is effective for various applications, including transforming coordinate systems and solving systems of equations.

An ellipse is a geometric shape characterized by its oval form and symmetric properties. The equation for an ellipse in standard form is given by: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

In our problem, we derived the expression \( 5x^2 + 3y^2 = 30\) from the matrix multiplication. To convert this into the standard form of an ellipse, we divided both sides of the equation by 30, resulting in: \( \frac{5x^2}{30} + \frac{3y^2}{30} = 1\). This simplifies to: \( \frac{x^2}{6} + \frac{y^2}{10} = 1\), which is the equation of an ellipse.

Here, \( a^2 = 6\) and \( b^2 = 10\), making \( a = \frac{\text{3}}{\text{sqrt(2)}}\) and \( b = \frac{\text{sqrt(2)}}{\text{5}}\). The ellipse's axes lengths are determined by these values. Understanding how to manipulate and transform such equations is crucial for analyzing elliptical shapes in coordinate geometry.

Algebraic proof involves demonstrating the validity of a mathematical statement using algebraic principles and logical reasoning. In this problem, our goal was to prove that the given matrix equation represents an ellipse.

We started by performing matrix multiplication in steps. First, we computed the product of the 2x2 and 2x1 matrices: \( \begin{array}{rr}5 & -7\7 & 3\right)\) and \( \begin{array}{l}x\y\right)\), resulting in \( \begin{array}{c}5x - 7y\7x + 3y\right)\). Second, we multiplied this result by \( \begin{array}{ll}x & y\ \right)\) to obtain the scalar expression \( 5x^2 + 3y^2\).

Upon establishing \( 5x^2 + 3y^2 = 30\), we compared this with the standard ellipse form, \( \frac{5x^2}{30} + \frac{3y^2}{30} = 1\), to confirm the ellipse representation. Such algebraic proofs are important for verifying mathematical concepts, ensuring correctness, and enhancing understanding of foundational principles in mathematics.